NUMBER SYSTEM
1. BASIC FORMULAE
1. (a+b)2=a2+b2+2ab
2. (a−b)2=a2+b2−2ab
3. (a+b)2−(a−b)2=4ab
4. (a+b)2+(a−b)2=2(a2+b2 )
5. (a2−b2)=(a+b)(a−b)
6. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
7. (a3+b3,)=(a+b)(a2−ab+b2)
8. (a3−b3)=(a−b)(a2+ab+b2)
9. (a3+b3+c3−3abc)=(a+b+c)(a2+b2+c2−ab−bc−ca)
10. If a+b+c=0, then a3+b3+c3,=3abc.
2. TYPES OF NUMBERS
I. Natural Numbers
Counting numbers 1,2,3,4,5,... are called natural numbers
II. Whole Numbers
All counting numbers together with zero form the set of whole numbers. Thus,
(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.
III. Integers
All natural numbers, 0 and negatives of counting numbers i.e., ...,−3,−2,−1,0,1,2,3,..... together form the set of integers.
(i) Positive Integers: 1,2,3,4,..... is the set of all positive integers.
(ii) Negative Integers: −1,−2,−3,..... is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.
IV. Even Numbers
A number divisible by 2 is called an even number, e.g.,2,4,6,8, etc.
V. Odd Numbers
A number not divisible by 2 is called an odd number. e.g.,1,3,5,7,9,11, etc.
VI. Prime Numbers
A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself. Prime numbers up to 100 are :2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
VII. Composite Numbers
Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4,6,8,9,10,12.
VII. Co-primes
Two natural numbers are said to be co-primes. If their HCF is 1.
Ex. (2,3), (7,8), (5,11) etc.
IX. Rational numbers
A number that can be expressed as p/q is called a rational number, where p and q are integers and q ≠ 0Ex. 3/5, 2/9, 11/12 etc.
X. Irrational number
A number that can not be expressed in the form of p/q is called a irrational number, where p and q are integers and q ≠ 0Ex. √2,√3, √5 etc
XI. Real numbers
Real numbers include both rational and irrational numbers.
3. TESTS OF DIVISIBILITY
1. Divisibility By 2
A number is divisible by 2, if its unit's digit is any of 0,2,4,6,8.
Example:
84932 is divisible by 2, while 65935 is not.
2. Divisibility By 3
A number is divisible by 3, if the sum of its digits is divisible by 3.
Example:
592482 is divisible by 3, since sum of its digits =(5+9+2+4+8+2)=30, which is divisible by 3. But, 864329 is not divisible by 3, since sum of its digits =(8+6+4+3+2+9)=32, which is not divisible by 3.
3. Divisibility By 4
A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
Example:
892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4. But, 749282 isnot divisible by 4, since the number formed by the last two digits is 82, which is not divisible by 4.
4. Divisibility By 5
A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
5. Divisibility By 6
A number is divisible by 6, if it is divisible by both 2 and 3.
Example:
The number 35256 is clearly divisible by 2.Sum of its digits =(3+5+2+5+6)=21, which is divisible by 3. Thus, 35256 isdivisible by 2 as well as 3. Hence, 35256 is divisible by 6.
6. Divisibility By 8
A number is divisible by 8, if the number formed by the last Three digits of the given number is divisible by 8.
Example:
953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8. But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.
7. Divisibility By 9
A number is divisible by 9, if the sum of its digits is divisible by 9.
Example:
60732 is divisible by 9, since sum of digits =(6+0+7+3+2)=18, which is divisible by 9. But, 68956 is not divisible by 9, since sum of digits =(6+8+9+5+6)=34, which is not divisible by 9.
8. Divisibility By 10
A number is divisible by 10, if it ends with 0.
Example:
96410, 10480 are divisible by 10, while 96375 is not.
9. Divisibility By 11
A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at evenplaces, is either 0 or a number divisible by 11.
Example:
The number 4832718 is divisible by 11, since :(sum of digits at odd places) - (sum of digits at even places) = =(8+7+3+4)−(1+2+8)=11, which is divisible by 11.
10. Divisibility By 12
A number is divisible by 12, if it is divisible by both 4 and 3.
Example:
Consider the number 34632. (i) The number formed by last two digits is 32, which is divisible by 4,(ii) Sum of digits =(3+4+6+3+2)=18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 isdivisible by 12.
R11. Divisibility By 14
A number is divisible by 14, if it is divisible by 2 as well as 7.
12. Divisibility By 15
A number is divisible by 15, if it is divisible by both 3 and 5. 13.
Divisibility By 16
A number is divisible by 16, if the number formed by the last4 digits is divisible by 16.
Example:
7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.
14.Divisibility by 17
Negative Oscillator for 17 is 5. The working for this is the same as in the case 7. Eg: check the divisibility of 1904 by 17
Sol: 1904 : 190 - 5 x 4 = 170, Since 170 is divisible by 17, the given number is also divisible by 17.
E.g 2: 957508 by 17 So1:957508: 95750 - 5 x 8 = 95710
95710 : 9571 - 5 x 0 = 9571 9571 : 957 - 5 x 1 = 952
952 : 95 - 5x2 =85 Since 85 is divisible by 17, the given number is divisible by 17.
15. Divisibility by 18
Any number which is a divisible by 9 has its last digit (unit-digit) even or zero, is divisible by 18. Eg. 926568 : Digit - Sum is a multiple of nine (i.e, divisible by 9) and unit digit (8) is even, hence the number is divisible by 18.
16. Divisibility by 19
If recall, the 'one-more' osculator for 19 is 2. The method is similar to that of 13, which is well known to us. Eg. 1 4 9 2 6 4 19/9/12/11/14
General rules of divisibility for all numbers:
- If a number is divisible by another number, then it is also divisible by all the factors of the other number.
- If two numbers are divisible by another number, then their sum and difference is also divisible by the other number.
- If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-primenumbers.
4. PROGRESSION
A succession of numbers formed and arranged in a definite order according to certain definite rule, is called aprogression.
1. Arithmetic Progression (A.P.)
If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P. An A.P. with first term a and common difference d is given by a,(a+d),(a+2d),(a+3d),.....
The nth term of this A.P. is given by Tn=a(n−1)d.
The sum of n terms of this A.P
Some Important Results:
- (1+2+3+....+n)=n(n+1)/2
- (l2+22+32+...+n2)=n(n+1 (2n+1)/6
- (13+23+33+...+n3)=[n(n+1)/2]2
2. Geometrical Progression (G.P.)
A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometricalprogression. The constant ratio is called the common ratio of the G.P.
A G.P. with first term a and common ratio r is :a,ar,ar2,.....
um of n terms, Sn=a(1−rn)/(1−r) when r < 1
NOTE
- (xm - am) is divisible by (x - a) for all values of m.
- (xm - am) is divisible by (x + a) for even values of m.
- (xm + am) is divisible by (x + a) for odd values of m.
EXAMPLES
1. If one-third of one-fourth of a number is 15, then three-tenth of that number is:
Sol - Let the number be x.
x
= 15 ⇒ x =180
2. The difference between a two-digit number and the number obtained by interchanging the positions of its digitsis 36. What is the difference between the two digits of that number?
Sol- Let the ten's digit be x and unit's digit be y.
Then, (10x + y) - (10y + x) = 36
⇒ 9(x - y) = 36
⇒ x - y = 4.
3. The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131.Their sum is:
Sol- Let the numbers be a, b and c.
Then, a2 + b2 + c2 = 138 and (ab + bc + ca) = 131.
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 138 + 2 x 131 = 400.
⇒ (a + b + c) = 400 = 20.
4. Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.
Sol-
⇒ x2 + 17x - 60 = 0
⇒ (x + 20)(x - 3) = 0
⇒ x = 3
5. The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is:
Sol-
xy = 120 and x2 + y2 = 289
∴ (x + y)2 = x2 + y2 + 2xy = 289 + (2 x 120) = 529
∴ x + y = √529 = 23.
6. What is the sum of two consecutive even numbers, the difference of whose squares is 84?
Sol-
Let the numbers be x and x + 2.Then, (x + 2)2 - x2 = 84
4x + 4 = 84
4x = 80
x = 20. The required sum = x + (x + 2) = 2x + 2 = 42.
7. It is being given that (232 + 1) is completely divisible by a whole number. Which of the following numbers iscompletely divisible by this number?
Sol-
Let 232 = x. Then, (232 + 1) = (x + 1).
Let (x + 1) be completely divisible by the natural number N. Then,
(296 + 1) = [(232)3 + 1] = (x3 + 1) = (x + 1)(x2 - x + 1), which is completely divisible by N, since (x + 1) is divisible by N.
8. What least number must be added to 1056, so that the sum is completely divisible by 23 ?
Sol-
On dividing 1056 by 23, remainder comes out 21
Required least number = 23-21 = 2
9. What is the unit digit in {(6374)1793 x (625)317 x (341491)}?
Sol-
Unit digit in (6374)1793 = Unit digit in (4)1793
= Unit digit in [(42)896 x 4]
= Unit digit in (6 x 4) = 4
Unit digit in (625)317 = Unit digit in (5)317 = 5
Unit digit in (341)491 = Unit digit in (1)491 = 1
Required digit = Unit digit in (4 ×5 ×1) = 0.
10. The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotientand the 15 as remainder. What is the smaller number ? Sol-
Sol-Let the smaller number be x. Then larger number = (x + 1365).
∴ x + 1365 = 6x + 15
⇒ 5x = 1350
⇒ x = 270
∴ Smaller number = 270
11. The sum of first 45 natural numbers is:
12.
Sol. = =
13.What will be remainder when (6767 + 67) is divided by 68 ?
Sol. (xn + 1) will be divisible by (x + 1) only when n is odd.
∴ (6767 + 1) will be divisible by (67 + 1)
∴ (6767 + 1) + 66, when divided by 68 will give 66 as remainder.
14. How many natural numbers are there between 23 and 100 which are exactly divisible by 6 ?
Sol-Required numbers are 24, 30, 36, 42, ..., 96
This is an A.P. in which a = 24, d = 6 and l = 96
Let the number of terms in it be n.
Then tn = 96 ⇒ a + (n - 1)d = 96
⇒ 24 + (n - 1) × 6 = 96
⇒ (n - 1) × 6 = 72
⇒ (n - 1) = 12
⇒ n = 13
Required number of numbers = 13.
15. On dividing a number by 357, we get 39 as remainder. On dividing the same number 17, what will be theremainder ?
Sol-Let x be the number and y be the quotient. Then,
x = 357 × y + 39
= (17 × 21 × y) + (17 ×2) + 5
= 17 × (21y + 2) + 5
∴ Required remainder = 5
16. In a division sum, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, what isthe dividend
Sol-divisor = 5×46 = 230
10×quotient = 230
Quotient = 23
Dividend = (divisor× quotient) + remainder
=(230 x 23) + 46
= 5290 + 46
= 5336.
17. (112 + 122 + 132 + ... + 202)
Sol-(112 + 122 + 132 + ... + 202) = (12 + 22 + 32 + ... + 202) - (12 + 22 + 32 + ... + 102)
= (2870 - 385)
= 2485
18. (xn - an) is completely divisible by (x - a), when
Sol- For every natural number n, (xn - an) is completely divisible by (x - a )